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Make a Flake: Exploring Symmetry with GeoGebra

Make a Flake: Exploring Symmetry with GeoGebra
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2012-11-29 01:33
PolygonTransformationGeometric Construction
In collections Transformations
This GeoGebra sketch lets users create snowflake art by constructing polygons and applying reflection and rotation symmetries. It demonstrates core concepts such as reflective symmetry, rotational symmetry, polygon construction, and geometric transformations. The construction uses GeoGebra commands Polygon, Ray, Point, and Mirror to generate objects like poly1, poly2, etc., and mirrors them to produce symmetric patterns. It is suitable for middle‑school and high‑school geometry lessons, as well, as STEAM activities that combine mathematics and art.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 3) → poly1, c, a, b, C
  2. Ray(A, C) → d
  3. Ray(A, B) → e
  4. Polygon(D, E, F, G, H) → poly2, d_1, e_1, f, g, h
  5. Polygon(E, I, J, K) → poly3, e_2, i, j, k
  6. Polygon(K, L, M, N, O, P) → poly4, k_1, l, m, n, o, p
  7. Polygon(G, Q, R) → poly5, r, g_1, q
  8. Point(d) → S
  9. Point(d) → T
  10. Point(e) → U
  11. Point(e) → V
  12. Polygon(S, T, U, V) → poly6, s, t, u, v
  13. Point(d) → W
  14. Point(e) → Z
  15. Point(e) → A_1
  16. Point(d) → C_1
  17. Polygon(W, Z, A_1, B_1, C_1) → poly7, w, z, a_1, b_1, c_1
  18. Mirror(S, d) → S'
  19. Mirror(T, d) → T'
  20. Mirror(U, d) → U'
  21. Mirror(V, d) → V'
  22. Polygon(S', T', U', V') → poly6', s', t', u', v'
  23. Mirror(W, d) → W'
  24. Mirror(Z, d) → Z'
  25. Mirror(A_1, d) → D_1
  26. Mirror(B_1, d) → E_1
  27. Mirror(C_1, d) → F_1
  28. Polygon(W', Z', D_1, E_1, F_1) → poly7', w', z', d_2, e_3, f_1
  29. Mirror(E, d) → E'
  30. Mirror(I, d) → I'
  31. Mirror(J, d) → J'
  32. Mirror(K, d) → K'
  33. Polygon(E', I', J', K') → poly3', e', i', j', k'
  34. Mirror(D, d) → D'
  35. Mirror(E, d) → E'_1
  36. Mirror(F, d) → F'
  37. Mirror(G, d) → G'
  38. Mirror(H, d) → H'
  39. Polygon(D', E'_1, F', G', H') → poly2', d', e'_1, f', g', h'
  40. Mirror(G, d) → G'_1
  41. Mirror(Q, d) → Q'
  42. Mirror(R, d) → R'
  43. Polygon(G'_1, Q', R') → poly5', r', g'_1, q'
  44. Mirror(K, d) → K'_1
  45. Mirror(L, d) → L'
  46. Mirror(M, d) → M'
  47. Mirror(N, d) → N'
  48. Mirror(O, d) → O'
  49. Mirror(P, d) → P'
  50. Polygon(K'_1, L', M', N', O', P') → poly4', k'_1, l', m', n', o', p'
  51. Mirror(A, d) → A'
  52. Mirror(B, d) → B'
  53. Ray(A', B') → e'_2
  54. Mirror(G'_1, e'_2) → G_1
  55. Mirror(Q', e'_2) → Q''
  56. Mirror(R', e'_2) → R''
  57. Polygon(G_1, Q'', R'') → poly5'', r'', g_2, q''
  58. Mirror(K'_1, e'_2) → H_1
  59. Mirror(L', e'_2) → L''
  60. Mirror(M', e'_2) → M''
  61. Mirror(N', e'_2) → N''
  62. Mirror(O', e'_2) → O''
  63. Mirror(P', e'_2) → P''
  64. Polygon(H_1, L'', M'', N'', O'', P'') → poly4'', h_1, l'', m'', n'', o'', p''
  65. Mirror(E', e'_2) → E''
  66. Mirror(I', e'_2) → I''
  67. Mirror(J', e'_2) → J''
  68. Mirror(K', e'_2) → K''
  69. Polygon(E'', I'', J'', K'') → poly3'', e'', i'', j'', k''
  70. Mirror(D', e'_2) → D''
  71. Mirror(E'_1, e'_2) → I_1
  72. Mirror(F', e'_2) → F''
  73. Mirror(G', e'_2) → G''
  74. Mirror(H', e'_2) → H''
  75. Polygon(D'', I_1, F'', G'', H'') → poly2'', d'', i_1, f'', g'', h''
  76. Mirror(W', e'_2) → W''
  77. Mirror(Z', e'_2) → Z''
  78. Mirror(D_1, e'_2) → J_1
  79. Mirror(E_1, e'_2) → K_1
  80. Mirror(F_1, e'_2) → L_1
  81. Polygon(W'', Z'', J_1, K_1, L_1) → poly7'', w'', z'', j_1, k_2, l_1
  82. Mirror(S', e'_2) → S''
  83. Mirror(T', e'_2) → T''
  84. Mirror(U', e'_2) → U''
  85. Mirror(V', e'_2) → V''
  86. Polygon(S'', T'', U'', V'') → poly6'', s'', t'', u'', v''
  87. Mirror(A, e'_2) → A'_1
  88. Mirror(C, e'_2) → C'
  89. Ray(A'_1, C') → d'_1
  90. Mirror(S'', d'_1) → S'''
  91. Mirror(T'', d'_1) → T'''
  92. Mirror(U'', d'_1) → U'''
  93. Mirror(V'', d'_1) → V'''
  94. Polygon(S''', T''', U''', V''') → poly6''', s''', t''', u''', v'''
  95. Mirror(W'', d'_1) → W'''
  96. Mirror(Z'', d'_1) → Z'''
  97. Mirror(J_1, d'_1) → M_1
  98. Mirror(K_1, d'_1) → N_1
  99. Mirror(L_1, d'_1) → O_1
  100. Polygon(W''', Z''', M_1, N_1, O_1) → poly7''', w''', z''', m_1, n_1, o_1
  101. Mirror(E'', d'_1) → E'''
  102. Mirror(I'', d'_1) → I'''
  103. Mirror(J'', d'_1) → J'''
  104. Mirror(K'', d'_1) → K'''
  105. Polygon(E''', I''', J''', K''') → poly3''', e''', i''', j''', k'''
  106. Mirror(D'', d'_1) → D'''
  107. Mirror(I_1, d'_1) → P_1
  108. Mirror(F'', d'_1) → F'''
  109. Mirror(G'', d'_1) → G'''
  110. Mirror(H'', d'_1) → H'''
  111. Polygon(D''', P_1, F''', G''', H''') → poly2''', d''', p_1, f''', g''', h'''
  112. Mirror(H_1, d'_1) → Q_1
  113. Mirror(L'', d'_1) → L'''
  114. Mirror(M'', d'_1) → M'''
  115. Mirror(N'', d'_1) → N'''
  116. Mirror(O'', d'_1) → O'''
  117. Mirror(P'', d'_1) → P'''
  118. Polygon(Q_1, L''', M''', N''', O''', P''') → poly4''', q_1, l''', m''', n''', o''', p'''
  119. Mirror(G_1, d'_1) → R_1
  120. Mirror(Q'', d'_1) → Q'''
  121. Mirror(R'', d'_1) → R'''
  122. Polygon(R_1, Q''', R''') → poly5''', r''', r_1, q'''
  123. Mirror(A', d'_1) → A''
  124. Mirror(B', d'_1) → B''
  125. Ray(A'', B'') → s_1
  126. Mirror(S''', s_1) → S_1
  127. Mirror(T''', s_1) → T_1
  128. Mirror(U''', s_1) → U_1
  129. Mirror(V''', s_1) → V_1
  130. Polygon(S_1, T_1, U_1, V_1) → poly8, s_2, t_1, u_1, v_1
  131. Mirror(W''', s_1) → W_1
  132. Mirror(Z''', s_1) → Z_1
  133. Mirror(M_1, s_1) → A_2
  134. Mirror(N_1, s_1) → B_2
  135. Mirror(O_1, s_1) → C_2
  136. Polygon(W_1, Z_1, A_2, B_2, C_2) → poly9, w_1, z_1, a_2, b_2, c_2
  137. Mirror(E''', s_1) → D_2
  138. Mirror(I''', s_1) → E_2
  139. Mirror(J''', s_1) → F_2
  140. Mirror(K''', s_1) → G_2
  141. Polygon(D_2, E_2, F_2, G_2) → poly10, d_3, e_4, f_2, g_3
  142. Mirror(D''', s_1) → H_2
  143. Mirror(P_1, s_1) → I_2
  144. Mirror(F''', s_1) → J_2
  145. Mirror(G''', s_1) → K_2
  146. Mirror(H''', s_1) → L_2
  147. Polygon(H_2, I_2, J_2, K_2, L_2) → poly11, h_2, i_2, j_2, k_3, l_2
  148. Mirror(R_1, s_1) → M_2
  149. Mirror(Q''', s_1) → N_2
  150. Mirror(R''', s_1) → O_2
  151. Polygon(M_2, N_2, O_2) → poly12, o_2, m_2, n_2
  152. Mirror(Q_1, s_1) → P_2
  153. Mirror(L''', s_1) → Q_2
  154. Mirror(M''', s_1) → R_2
  155. Mirror(N''', s_1) → S_2
  156. Mirror(O''', s_1) → T_2
  157. Mirror(P''', s_1) → U_2
  158. Polygon(P_2, Q_2, R_2, S_2, T_2, U_2) → poly13, p_2, q_2, r_2, s_3, t_2, u_2
  159. Mirror(S, e) → S'_1
  160. Mirror(T, e) → T'_1
  161. Mirror(U, e) → U'_1
  162. Mirror(V, e) → V'_1
  163. Polygon(S'_1, T'_1, U'_1, V'_1) → poly6'_1, s'_1, t'_1, u'_1, v'_1
  164. Mirror(W, e) → W'_1
  165. Mirror(Z, e) → Z'_1
  166. Mirror(A_1, e) → V_2
  167. Mirror(B_1, e) → W_2
  168. Mirror(C_1, e) → Z_2
  169. Polygon(W'_1, Z'_1, V_2, W_2, Z_2) → poly7'_1, w'_1, z'_1, v_2, w_2, z_2
  170. Mirror(E, e) → E'_2
  171. Mirror(I, e) → I'_1
  172. Mirror(J, e) → J'_1
  173. Mirror(K, e) → K'_2
  174. Polygon(E'_2, I'_1, J'_1, K'_2) → poly3'_1, e'_3, i'_1, j'_1, k'_2
  175. Mirror(K, e) → K'_3
  176. Mirror(L, e) → L'_1
  177. Mirror(M, e) → M'_1
  178. Mirror(N, e) → N'_1
  179. Mirror(O, e) → O'_1
  180. Mirror(P, e) → P'_1
  181. Polygon(K'_3, L'_1, M'_1, N'_1, O'_1, P'_1) → poly4'_1, k'_3, l'_1, m'_1, n'_1, o'_1, p'_1
  182. Mirror(G, e) → G'_2
  183. Mirror(Q, e) → Q'_1
  184. Mirror(R, e) → R'_1
  185. Polygon(G'_2, Q'_1, R'_1) → poly5'_1, r'_1, g'_2, q'_1
  186. Mirror(D, e) → D'_1
  187. Mirror(E, e) → E'_3
  188. Mirror(F, e) → F'_1
  189. Mirror(G, e) → G'_3
  190. Mirror(H, e) → H'_1
  191. Polygon(D'_1, E'_3, F'_1, G'_3, H'_1) → poly2'_1, d'_2, e'_4, f'_1, g'_3, h'_1